Abstract in another language

This thesis consists of two parts dealing with different so far unsolved problems in the field of pattern formation theory. The first part studies the effects of restricting pattern formation to a finite domain - a scenario that is omnipresent in nature. In the second part we identify and investigate a new phase separation phenomenon in active systems with a conservation law - the so-called active phase separation.

In a first publication we show that physical boundaries generically lead to a reflection effect for nonlinear traveling waves.This reflection forces systems that show traveling waves in large extended systems into a standing wave pattern if the system becomes sufficiently short.We also identify bands of stable standing waves with different numbers of nodes, allowing for transitions between different standing wave patterns.This generic result is especially relevant for the Min protein system that plays a crucial role in the cell division process of the bacterium E. coli. Thereby the Min proteins show a traveling wave pattern on large extended membranes in in vitro experiments, while inside a cell a standing wave-like pattern is observed.

Finite domains for patterns can also be generated without hard physical boundaries. Instead the control parameter that switches the system between a patterned state and a homogeneous state can be varied spatially in a way that it suppresses the pattern in one region and allows it in another. A possible experimental realization for this scenario are light-sensitive chemical reactions where the pattern formation process can be enhanced or inhibited using an illumination mask. We figure out that the steepness of the variation from a sub- to a supercritical control parameter influences the orientation of stripe patterns in two spatial dimensions. For steep step-like control parameter drops, the stripes favor a orientation parallel to the control parameter variation. For smooth ramp-like drops on the other hand, they favor a perpendicular orientation. This also implies that the orientation of stripes will switch from parallel to perpendicular when decreasing the steepness of the drop. This transition can be understood with the decreasing importance of local resonance effects induced by the control parameter drop.

In another way, a control parameter drop also influences traveling wave pattern in one dimension. While again local resonance effects are important, the control parameter drop there leads to four different wave patterns depending on the group velocity. For small group velocities, the traveling wave pattern thereby fills the whole supercritical domain forming a filled state. Increasing the group velocity will confine the pattern to one side of the supercritical domain.Even higher group velocities induce a state with a time-dependent amplitude of the wave pattern - a so-called blinking state. Thereby both left- and right-traveling waves occur whose amplitudes change periodically in time. Increasing the group velocity further leads to a return of a wave state with a stationary amplitude. In the counter-propagating wave state we find a left-moving wave in the left half of the supercritical domain and a right-moving wave in the right half.

The second part of this thesis attends to a phenomenon resembling phase separation or demixing in several systems with a conserved quantity. Among them are the assembly of proteins in different halves of a polarized cell, the aggregation of cell colonies that communicate chemotactically or clustering behavior of active Brownian particles. Even if these different systems constantly consume energy locally - rendering them non-equilibrium systems - all of them show a transition from a homogenous to a state with a dense and a dilute phase similar to classical phase separation.We are able to show that this similarity is indeed not coincidental. Instead, models of systems from very different fields can be mapped onto one universal equation close to the onset of the phase separation process. This equation turns out to be the Cahn-Hilliard equation - an equation that is usually used to describe phase separation in thermal equilibrium. We demonstrate that this equation is also the universal description of what we call active phase separation. In our publications we introduce a new kind of weakly nonlinear analysis that allows to directly link the parameters of the original system to those of the Cahn-Hilliard equation. This allows to confirm the validity of our approach by comparing numerical simulations of the different original systems to the corresponding Cahn-Hilliard model. We thereby find a convincing agreement in both stationary profiles, as well as the dynamical evolution of the two. We furthermore extend the weakly nonlinear analysis to the next higher order, which is especially interesting for active Brownian systems showing so-called motility-induced phase separation. In those systems the significance of higher order contributions is highly discussed. We are again able to directly map the original system to an extended Cahn-Hilliard model, which allows to identify straightforward the relevant contributions for a given model.